The aim of the proposed research is to provide a complete mathematical analysis of the dynamics of a closed population subject to an arbitrary sequence of age-specific fertility and mortality rates over time. Although the continuous model for the birth density will also be studied, primary attention will be given to the discrete model for the age distribution at time t, xt equals AtAt-1...A1x0, t equals 1,2,... We shall thus be most interested in investigating the behavior of sequences of products of population projection matrices. Our analysis will lead to the derivation of an expression linking age distribution to a "minimal past" history of vital rates; similar expressions can then be derived relating such functions of age distribution as population size, the dependency ratio, and crude vital rates to the "minimal past." To arrive at these expressions, we shall investigate successively limits of the process, convergence to these limits, and the relationship of period and cohort measures defined on the process. Special attention will be given in the analysis to hose types of change in vital rates which have occurred in the past, and to those which may be expected to hold in the future. The basic model will be expanded to include factors other than age, such as parity, marital status, and contraceptive usage, and the analysis of the basic model will be extended to these models.